Determinism, well-posedness, and applications of the ultrahyperbolic wave equation in spacekime
Spatiotemporal dynamics of many natural processes, such as elasticity, heat propagation, sound waves, and fluid flows are often modeled using partial differential equations (PDEs). Certain types of PDEs have closed-form analytical solutions, some permit only numerical solutions, some require appropr...
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| Format: | Article |
| Language: | English |
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Elsevier
2022-06-01
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| Series: | Partial Differential Equations in Applied Mathematics |
| Subjects: | |
| Online Access: | http://www.sciencedirect.com/science/article/pii/S2666818122000122 |
| _version_ | 1811241005436895232 |
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| author | Yuxin Wang Yueyang Shen Daxuan Deng Ivo D. Dinov |
| author_facet | Yuxin Wang Yueyang Shen Daxuan Deng Ivo D. Dinov |
| author_sort | Yuxin Wang |
| collection | DOAJ |
| description | Spatiotemporal dynamics of many natural processes, such as elasticity, heat propagation, sound waves, and fluid flows are often modeled using partial differential equations (PDEs). Certain types of PDEs have closed-form analytical solutions, some permit only numerical solutions, some require appropriate initial and boundary conditions, and others may not have stable, global, or even well-posed solutions. In this paper, we focus on one-specific type of second-order PDE — the ultrahyperbolic wave equation in multiple time dimensions. We demonstrate the wave equation solutions in complex time (kime) and show examples of the Cauchy initial value problem in space-kime.We extend the classical formulation of the dynamics of the wave equation with respect to positive real longitudinal time. The solutions to the Cauchy boundary value problem in multiple time dimensions are derived in Cartesian, polar, and spherical coordinates. These include both bounded and unbounded spatial domains. Some example solutions are shown in the main text with additional web-based dynamic illustrations of the wave equation solutions in space-kime shown in the appendix. Solving PDEs in complex time has direct connections to data science, where solving under-determined linear modeling problems or specifying the initial conditions on limited spatial dimensions may be insufficient to forecast, classify, or predict a prospective value of a parameter or a statistical model. This approach extends the notion of data observations, anchored at ordered longitudinal events, to complex time, where observables need not follow a strict positive-real structural arrangement, but instead could traverse the entire kime plane. |
| first_indexed | 2024-04-12T13:29:29Z |
| format | Article |
| id | doaj.art-8a3c61c2f2f04097bee0a26953481086 |
| institution | Directory Open Access Journal |
| issn | 2666-8181 |
| language | English |
| last_indexed | 2024-04-12T13:29:29Z |
| publishDate | 2022-06-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Partial Differential Equations in Applied Mathematics |
| spelling | doaj.art-8a3c61c2f2f04097bee0a269534810862022-12-22T03:31:13ZengElsevierPartial Differential Equations in Applied Mathematics2666-81812022-06-015100280Determinism, well-posedness, and applications of the ultrahyperbolic wave equation in spacekimeYuxin Wang0Yueyang Shen1Daxuan Deng2Ivo D. Dinov3Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA; Statistics Online Computational Resource, Department of Health Behavior and Biological Sciences, University of Michigan, Ann Arbor, MI 48109, USAElectrical Computer Engineering Division, University of Michigan, Ann Arbor, MI 48109, USA; Statistics Online Computational Resource, Department of Health Behavior and Biological Sciences, University of Michigan, Ann Arbor, MI 48109, USADepartment of Statistics, University of Michigan, Ann Arbor, MI 48109, USA; Statistics Online Computational Resource, Department of Health Behavior and Biological Sciences, University of Michigan, Ann Arbor, MI 48109, USAStatistics Online Computational Resource, Department of Health Behavior and Biological Sciences, University of Michigan, Ann Arbor, MI 48109, USA; Department of Computational Medicine and Bioinformatics, University of Michigan, Ann Arbor, MI 48109, USA; Correspondence to: Statistics Online Computational Resource, University of Michigan, 426 N. Ingalls Str., Ann Arbor, MI 48109, USA.Spatiotemporal dynamics of many natural processes, such as elasticity, heat propagation, sound waves, and fluid flows are often modeled using partial differential equations (PDEs). Certain types of PDEs have closed-form analytical solutions, some permit only numerical solutions, some require appropriate initial and boundary conditions, and others may not have stable, global, or even well-posed solutions. In this paper, we focus on one-specific type of second-order PDE — the ultrahyperbolic wave equation in multiple time dimensions. We demonstrate the wave equation solutions in complex time (kime) and show examples of the Cauchy initial value problem in space-kime.We extend the classical formulation of the dynamics of the wave equation with respect to positive real longitudinal time. The solutions to the Cauchy boundary value problem in multiple time dimensions are derived in Cartesian, polar, and spherical coordinates. These include both bounded and unbounded spatial domains. Some example solutions are shown in the main text with additional web-based dynamic illustrations of the wave equation solutions in space-kime shown in the appendix. Solving PDEs in complex time has direct connections to data science, where solving under-determined linear modeling problems or specifying the initial conditions on limited spatial dimensions may be insufficient to forecast, classify, or predict a prospective value of a parameter or a statistical model. This approach extends the notion of data observations, anchored at ordered longitudinal events, to complex time, where observables need not follow a strict positive-real structural arrangement, but instead could traverse the entire kime plane.http://www.sciencedirect.com/science/article/pii/S2666818122000122PDEsStatisticsData scienceComplex timeKimeSpacekime analytics |
| spellingShingle | Yuxin Wang Yueyang Shen Daxuan Deng Ivo D. Dinov Determinism, well-posedness, and applications of the ultrahyperbolic wave equation in spacekime Partial Differential Equations in Applied Mathematics PDEs Statistics Data science Complex time Kime Spacekime analytics |
| title | Determinism, well-posedness, and applications of the ultrahyperbolic wave equation in spacekime |
| title_full | Determinism, well-posedness, and applications of the ultrahyperbolic wave equation in spacekime |
| title_fullStr | Determinism, well-posedness, and applications of the ultrahyperbolic wave equation in spacekime |
| title_full_unstemmed | Determinism, well-posedness, and applications of the ultrahyperbolic wave equation in spacekime |
| title_short | Determinism, well-posedness, and applications of the ultrahyperbolic wave equation in spacekime |
| title_sort | determinism well posedness and applications of the ultrahyperbolic wave equation in spacekime |
| topic | PDEs Statistics Data science Complex time Kime Spacekime analytics |
| url | http://www.sciencedirect.com/science/article/pii/S2666818122000122 |
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